This is a mix of continuous and batch processes.
The batch process is refilling the tank.
The continuous process is loss of water in the cooling tower by evaporation. The issue is that literally ideally only
water evaporates to become vapor, which vapor exits the process through the top of the tower, motivated by the fan, while the
solids that were dissolved in the evaporated water do not evaporate (sublimate, much) and instead end up in the remaining cooled un-evaporated water. That cooled water cycles back and forth between the refrigeration process, picking up heat to start the process over again.
In the meantime, fresh water is added to the process to replace the water that evaporated; I think this is where the buffer tank comes in. The fresh water has dissolved solids, and even though those solids are at a lower concentration than the water evaporation, there is no place for those solids (yet) to go once they get to the heating/cooling circuit. The volume of water in the circuit is more or less constant, as any volume lost through evaporation is made up. Since the total amount of solids in the circuit increases over time while the volume is constant, the concentration would increase. The only way to deal with this is to remove some of the water (called blowdown in the steam turbine process), which as noted has a higher solids concentration, from the circuit. Since this will be made up by adding an equal volume of fresh water with a lower solids concentration, the blowdown itself represents a loss of solids in the system, which will eventually exactly match the gain of solids from the fresh water replacing volume lost both from evaporation and from blowdown.
Conductance is used as proxy for the dissolved solids concentration, specifically for dissolved ions concentration, a.k.a. salinity.
It's not clear if
- EITHER the buffer tank is part of that cycle,
- i.e. if the water cycles from the refrigeration unit, to the cooling tower, to the buffer tank, then back to refrigeration, etc.,
- OR resupplying makeup water from the buffer tank to the refrigeration/tower circuit is configured as an external makeup.
- i.e. the water cycles from the refrigeration unit, to the cooling tower, then back to refrigeration, etc.,
- while the buffer tank supplies water, perhaps to maintain level in a sump at the bottom of the cooling tower to prevent pumps from being starved.
Either way, I don't think it changes the math, which boils (
) down to two material balances:
- net water volume accumulation = in - out = makeup - (evaporation + blowdown)
- evaporation is the liquid volume lost to evaporative cooling
- net solids accumulation = in - out = (makeup * ECmakeup) - (blowdown * ECblowdown)
- EC is Electrical Conductance (the siemens measurement), assumed to be a proxy for dissolved solids i.e. EC=0 when dissolved solids are absent, and EC value is linear with the concentration of solids.
- N.B. this assumes there is no loss from evaporation or sublimation, but there could be some from entrained water droplets.
Both accumulations will be 0 for continuous operation; EC
makeup is known (not really, but more later), evaporation is known, EC
blowdown has a maximum spec and so is more or less known. That leaves two unknowns: makeup volume and blowdown volume.
That is two equations with two independent quantities, makeup and blowdown, which can be solved simply via substitution:
- blowdown = evaporation * ECmakeup) / (ECblowdown- ECmakeup)
- makeup = evaporation + blowdown
Sanity checks:
- Using makeup water with a lower ECmakeup decreases the numerator and increases the denominator in equation 1, so it decreases blowdown, which decreases makeup in equation 2,
- and vice versa.
Also note that increasing the EC
blowdown target spec increases the denominator in equation 1, which in turn decreases blowdown and makeup, but may also have long-term maintenance costs because the concentration of solids in the circulating system is higher.
That's the easy part; optimizing EC
makeup to minimize operating cost based on the cost of various blends of the three water types is the interesting part. It's mostly bookkeeping (volume and mass balance, analogous to the equations above, plus a cost function that is also linear) as blending ECs is linear and costs are linear, so there should be a straightforward path to the solution.
As a start, it should be easy to calculate estimated relative costs for using each of the three water types alone, then introduce a few percent of the other types into each of those and see if the costs go up or down. Since this problem is mostly linear or at least monotonic (that denominator is a burr under the saddle), I suspect that using only one water type will be the optimum choice. But if any of the costs go down when blending, then increase the introduced blends and try again i.e. follow steepest descent.
This is also a small enough problem that simple brute force modeling of all possible blends in 1% or 10% increments is manageable. It could even be done in eXcel (and I can't believe I am suggesting that
).
